2. Free-free emission from a wind
where is the mass loss rate, is the mass density, v is the wind velocity, is the hydrogen mass, is the ion density and µ is the mean atomic weight per ion. Because the observed emission originates from the ionized gas component in the wind it is necessary to consider the ionization balance in the wind. Let be the electron density, the ion density of the ions from the element with atomic number Z and with a charge z, the total density of neutrals and ions of the element with atomic number Z and the element abundance. Note that . The electron density is given by
while the ion density is given by
The average number of electrons per ion is given by and the mean atomic mass per ion is given by
The fraction of neutral hydrogen is given by .
Since there is no information available from observations concerning the ionization balance in the wind, the most plausible assumption one can make is that, after an initial acceleration of the wind, the ionization balance is frozen-in due to the rapid decrease of the density. The ionization balance in the wind then reflects the ionization balance at the base of the wind which is determined by collisional ionization equilibrium. We determined the values of , , , µ and using the ionization balance by Arnaud and Raymond (1992) for iron and by Arnaud and Rothenflug (1985) for the other elements. Solar photospheric abundances were assumed (Anders and Grevesse, 1989). The results are presented in Table 1 for a number of wind temperatures in the range .
The linear free-free absorption coefficient is given by (Allen, 1973)
For calculating the gaunt factors we followed the procedure outlined by Waters and Lamers (1984) with the exception that Lamers and Waters write with the mean value of the squared atomic charge. In our case separating out would be impractical given the form of Eq. (6). For low values of we use the expression for the gaunt factor given by Allen (1973) but with the correction discussed by Leitherer and Robert (1991). For high values of we use the expression by Mewe et al. (1986) (see also Gronenschild and Mewe, 1978). In practice we took the maximum value for , at a given temperature and frequency, which resulted from these approximations. The results are shown in Fig. 2.
By introducing the dimensionless parameters and , the absorption coefficient can be written as
The constant is independent of the frequency.
The total flux emitted by the star and the wind is easily calculated following the standard procedure outlined by Wright and Barlow (1975), Panagia and Felli (1975) and Lamers and Waters (1984). The total flux is given by
where , , is the Planck function, is the black-body temperature of the star, is the wind temperature and q is the impact parameter of the line of sight (see e.g. Fig. 1 in Lamers and Waters, 1984). The optical depth at impact parameter q is given by
The optical depth depends of course on the assumed velocity law of the wind for which no information is available from observations. Therefore we use in this paper a velocity law for a wind which is accelerating up to some distance and then obtains its final velocity
The acceleration of the wind is determined by the value of . The dimensionless velocity is given by for and for .
Given a velocity law for the wind, the run of the ion density in the wind can be determined from Eq. (1). In Table 2 the related expressions for are listed. Also listed in Table 2 are the expressions for the emission measure EM of the wind and the neutral hydrogen column density along the line of sight at the centre of the star. For the emission measure we took into account that part of the wind does not contribute since it is obscured by the star.
Table 2. Expressions for: the optical depth at impact parameter q (see Eqs. (10) and (11)), the emission measure EM of the visible part of a isothermal stellar wind, the neutral hydrogen column density , the effective radius and the effective optical depth .
With the expressions for , as given in Table 2, part of Eq. (10) can be evaluated analytically resulting in
with and the incomplete Gamma function (Abramowitz and Stegun, 1968). The first term on the right accounts for the emission by the star, which can be attenuated by absorption due to the wind. The second term accounts for the emission from the cone in front of the star while the third term accounts for the emission from the wind acceleration region outside the previously mentioned cone. The fourth term described the emission from the volume (outside the cone) where the wind has reached its terminal velocity. Note that for a wind with constant velocity () the third term does not contribute. At low frequencies, at which is large, only the last term effectively contributes to the flux resulting in
which is identical to the expression found by Wright and Barlow (1975). At higher frequencies, at which becomes small, we can make the approximation in Eqs. (10) and (11). Evaluation of the resulting integrals shows that, at frequencies at which the emission is optically thin, the flux is given by
The last identity in Eq. (13) results from applying Kirchhoff's law. As could be anticipated, in the optically thin part of the spectrum the flux is composed of the contribution by the star and the free-free emission from a wind with an emission measure EM.
For completeness we present in Table 2 also the effective radius and the effective optical depth (see Wright and Barlow, 1975). The effective radius is defined by assuming that the emission at a given frequency originates from the volume at
with given by Eq. (10). The effective optical depth is then defined by .
© European Southern Observatory (ESO) 1997
Online publication: July 3, 1998